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prove or disprove following statements, explaining your reasoning. prove or disprove questions : total 7 questions

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4. (5 points) PROVE or DISPROVE the following statement.

There exist distinct rational numbers a and b such that (a – 1)(b − 1) = 1

5. (5 points) Suppose that A = {*,2,Ø} and B = {{0},{1,2}}. Let f : A + B be the function defined

by f(*) = {@}, f(2) = {0}, and f(0) = {1,2}. Determine whether f is one-to-one, onto, both, or

neither. Explain your answer.

PART III: PROOFS

Instructions: Prove the following statements, explaining your reasoning. Partial credit will be awarded for

an incorrect proof that shows progress towards a correct proof.

You must complete problem 1, and three of problems 2-5. That is, you will skip one of the problems 2-5.

Draw a large X through the problem you do not wish to be graded.

1. (20 points) Using mathematical induction, prove that for all positive integers n,

11 +19+27+ … + (8n +3) = n(4n+7)

2. (10 points) Prove: For all sets A, B and C,

A-(BUC) = (A – B) n(A-C).

3. (10 points) Suppose a and b are integers. Prove that if a +b is even, then a – b is also even.

4. (10 points) Prove that there exists no positive integer n such that 2n < n2
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Tags:
Mathematical induction
rational numbers
Division Algorithm
Induction hypothesis
Positive integrers
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